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Introduction

In mathematics, functions are used to describe the relationship between variables. A function is a rule that assigns to each input value, or input, exactly one output value, or output. In this article, we will explore the function h(x) defined by the rule h(x) = (1/5)^x and find its values for each x-value in a given table.

The Function h(x)

The function h(x) is defined as h(x) = (1/5)^x. This function is an exponential function, where the base is 1/5 and the exponent is x. The function h(x) can be evaluated for any real number x.

Evaluating h(x) for Different Values of x

To evaluate h(x) for different values of x, we can substitute the value of x into the function h(x) = (1/5)^x. Let's consider the values of x in the table.

Table of Values

x h(x)
-2
0
1
2
3

Finding h(x) for Each Value of x

h(-2)

To find h(-2), we substitute x = -2 into the function h(x) = (1/5)^x.

h(-2) = (1/5)^(-2) = 5^2 = 25

h(0)

To find h(0), we substitute x = 0 into the function h(x) = (1/5)^x.

h(0) = (1/5)^0 = 1

h(1)

To find h(1), we substitute x = 1 into the function h(x) = (1/5)^x.

h(1) = (1/5)^1 = 1/5

h(2)

To find h(2), we substitute x = 2 into the function h(x) = (1/5)^x.

h(2) = (1/5)^2 = 1/25

h(3)

To find h(3), we substitute x = 3 into the function h(x) = (1/5)^x.

h(3) = (1/5)^3 = 1/125

Conclusion

In this article, we defined the function h(x) = (1/5)^x and evaluated its values for each x-value in a given table. We found that h(-2) = 25, h(0) = 1, h(1) = 1/5, h(2) = 1/25, and h(3) = 1/125. The function h(x) is an exponential function, where the base is 1/5 and the exponent is x. This function can be used to model real-world phenomena, such as population growth and decay.

Applications of h(x)

The function h(x) has several applications in mathematics and other fields. Some of these applications include:

  • Population growth and decay: The function h(x) can be used to model population growth and decay. For example, if the initial population is 100 and the growth rate is 1/5, then the population after x years can be modeled by h(x) = (1/5)^x * 100.
  • Finance: The function h(x) can be used to calculate the future value of an investment. For example, if an investment grows at a rate of 1/5 per year, then the future value of the investment after x years can be modeled by h(x) = (1/5)^x * P, where P is the initial investment.
  • Science: The function h(x) can be used to model the decay of radioactive materials. For example, if the half-life of a radioactive material is 10 years, then the amount of the material remaining after x years can be modeled by h(x) = (1/2)^x * A, where A is the initial amount of the material.

Conclusion

Introduction

In our previous article, we explored the function h(x) defined by the rule h(x) = (1/5)^x and found its values for each x-value in a given table. In this article, we will answer some frequently asked questions about the function h(x) and its applications.

Q: What is the domain of the function h(x)?

A: The domain of the function h(x) is all real numbers, since the exponent x can be any real number.

Q: What is the range of the function h(x)?

A: The range of the function h(x) is all positive real numbers, since the base 1/5 is always positive and the exponent x can be any real number.

Q: How do I evaluate h(x) for a given value of x?

A: To evaluate h(x) for a given value of x, you can substitute the value of x into the function h(x) = (1/5)^x. For example, to find h(2), you would substitute x = 2 into the function and get h(2) = (1/5)^2 = 1/25.

Q: Can I use h(x) to model population growth and decay?

A: Yes, you can use h(x) to model population growth and decay. For example, if the initial population is 100 and the growth rate is 1/5, then the population after x years can be modeled by h(x) = (1/5)^x * 100.

Q: Can I use h(x) to calculate the future value of an investment?

A: Yes, you can use h(x) to calculate the future value of an investment. For example, if an investment grows at a rate of 1/5 per year, then the future value of the investment after x years can be modeled by h(x) = (1/5)^x * P, where P is the initial investment.

Q: Can I use h(x) to model the decay of radioactive materials?

A: Yes, you can use h(x) to model the decay of radioactive materials. For example, if the half-life of a radioactive material is 10 years, then the amount of the material remaining after x years can be modeled by h(x) = (1/2)^x * A, where A is the initial amount of the material.

Q: What are some real-world applications of the function h(x)?

A: Some real-world applications of the function h(x) include:

  • Population growth and decay: h(x) can be used to model population growth and decay in biology and ecology.
  • Finance: h(x) can be used to calculate the future value of an investment in finance.
  • Science: h(x) can be used to model the decay of radioactive materials in physics and chemistry.
  • Engineering: h(x) can be used to model the behavior of electrical circuits and electronic systems.

Conclusion

In conclusion, the function h(x) = (1/5)^x is a powerful tool that can be used to model real-world phenomena in a variety of fields. By understanding the properties and applications of h(x), you can use it to make predictions and model complex systems.

Additional Resources

For more information on the function h(x) and its applications, please see the following resources:

  • Textbooks: "Calculus" by Michael Spivak, "Mathematics for Computer Science" by Eric Lehman and Tom Leighton.
  • Online resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha.
  • Software: Mathematica, Maple, MATLAB.

Glossary

  • Exponential function: A function of the form f(x) = a^x, where a is a positive real number.
  • Domain: The set of all possible input values for a function.
  • Range: The set of all possible output values for a function.
  • Population growth and decay: The increase or decrease in the size of a population over time.
  • Finance: The study of the management of money and investments.
  • Science: The study of the natural world through observation and experimentation.
  • Engineering: The application of scientific and mathematical principles to design and develop solutions to real-world problems.